12.215 Modern Navigation

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12.215 Modern Navigation

• Thomas Herring (tah_at_mit.edu),
• MW 1100-1230 Room 54-322
• http//geoweb.mit.edu/tah/12.215

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Summary of Last Class

• GPS measurements
• Basics of pseudorange measurements
• Phase measurements (allow millimeter level
  position with GPS and cm in real-time)
• Examine some GPS data.
• Positioning modes
• Dilution of precision numbers

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Todays Class

• Atmospheric propagation.
• Basic structure of the atmosphere
• Refractive index of atmospheric air
• Accounting for atmospheric delays in precise GPS
  navigation
• Hydrostatic and wet delays
• Use of GPS for weather forecasting

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Basic atmospheric structure
Troposphere is where the temperature stops
decreasing in the atmosphere. (10 km altitude)
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Troposhere

• Lots of examples of web-based documents about the
  atmosphere See for example.
• http//www-das.uwyo.edu/geerts/cwx/notes/chap01/t
  ropo.html
• Tropopause is where temperature stops decreasing.
  Generally at pressure levels of about 300 mbar
  but can be as low as 500 mbar.
• Sometimes term tropospheric delay used but this
  is only about 70 of delay.
• Generally by height of 50-100km all of
  atmospheric delay accounted for.
• Troposphere is where weather system occur and
  aircraft fly on the tropopause.

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Refractivity of air

• Air is made up of specific combination of gases,
  the most important ones being oxygen and
  nitrogen.
• Each gas has its own refractive index that
  depends on pressure and temperature.
• For the main air constituents, the mixing ratio
  of the constituents is constant and so the
  refractivity of a packet of air at a specific
  pressure and temperature can be defined.
• The one exception to this is water vapor which
  has a very variable mixing ratio.
• Water vapor refractivity also depends on
  density/temperature due to dipole component.

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Refractivity of air

• The refractivity of moist air is given by
• For most constituents, refractivity depends on
  density (ie., number of air molecules). Water
  vapor dipole terms depends on temperature as well
  as density

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Refractivity in terms of density

• We can write the refractivity in terms of
  density
• Density r is the density of the air parcel
  including water vapor. R is universal gas
  constant, Md and Mw are molecular weights. Zw is
  compressibility (deviation from ideal gas law)
  See Davis, J. L., T. A. Herring, and I.I.
  Shapiro, Effects of atmospheric modeling errors
  on determinations of baseline vectors from VLBI,
  J. Geophys. Res., 96, 643650, 1991.

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Integration of Refractivity

• To model the atmospheric delay, we express the
  atmospheric delay as
• Where the atm path is along the curved
  propagation path vac is straight vacuum path, z
  is height for station height Z and m(e) is a
  mapping function. (Extended later for
  non-azimuthally symmetric atmosphere)
• The final integral is referred to as the zenith
  delay

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Zenith delay

• The zenith delay is determined by the integration
  of refractivity vertically.
• The atmospheric is very close to hydrostatic
  equilibrium meaning that surface pressure is
  given by the vertical integration of density.
  Since the first term in refractivity depends only
  on density, its vertical integration will depend
  only on surface pressure. This integral is
  called the zenith hydrostatic delay (ZHD).
  (Often referred to as dry delay but this is
  incorrect because has water vapor contribution).

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Zenith hydrostatic delay

• The Zenith hydrostatic delay is given by
• Where gm is mean value of gravity in column of
  air (Davis et al. 1991)gm9.8062(1-0.00265cos(2f)
  -3.1x10-7(0.9Z7300)) ms-2
• Ps is total surface pressure (again water vapor
  contribution included)
• Since Ps is 1013 mbar at mean sea level typical
  ZHD 2.3 meters

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Zenith wet delay

• The water vapor delay (second term in
  refractivity) is not so easily integrated because
  of distribution of water vapor with height.
• Surface measurements of water vapor pressure
  (deduced from relative humidity) are not very
  effective because it can be dry at surface and
  moist above and visa versa.
• Only effective method is to sense the whole
  column of water vapor. Can be done with water
  vapor radiometer (WVR) which infers water vapor
  delay from thermal emission from water vapor
  molecules and some laser profiling methods
  (LIDAR). Both methods are very expensive (200K
  per site)

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Zenith wet delay

• In meteorology, the term Precipitable water
  (PW) is used. This is the integral of water
  vapor density with height and equals the depth of
  water if all the water vapor precipitated as rain
  (amount measured on rain gauge).
• If the mean temperature of atmosphere is known,
  PW can be related to Zenith Wet Delay (ZWD) (See
  next page)

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PW and ZWD

• Relationship
• The factor for conversion is 6.7 mm delay/mm PW
• This relationship is the basis of ground based
  GPS meteorology where GPS data are used to
  determine water vapor content of atmosphere.
• ZWD is usually between 0-30cm.

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Mapping functions

• Zenith delays discussed so far how to relate to
  measurements not at zenith
• Problem has been studied since 1970s.
• In simplest form, for a plain atmosphere,
  elevation angle dependence would behave as
  1/sin(elev). (At the horizon, elev0 and this
  form goes to infinity.
• For a spherically symmetric atmosphere, the
  1/sin(elev) term is tempered by curvature
  effects.
• Most complete form is continued fraction
  representation (Davis et al., 1991).

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Continued fraction mapping function

• Basic form of mapping function was deduced by
  Marini (1972) and matches the behavior of the
  atmosphere at near-zenith and low elevation
  angles. Form is

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Truncated version

• When the mapping function is truncated to the
  finite number of terms then the form is

Davis et al. 1991 solved problem by using tan for
second sin
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Mapping functions

• Basic problem with forming a mapping function is
  determining the coefficient a,b, c etc for
  specific weather conditions.
• There are different parameterizations
• Niell mapping function uses a, b,c that are
  latitude, height and time of year dependent
• MTT (MIT Temperature) model uses temperature as
  proxy for atmospheric conditions.
• Recent Niell work uses height of 500mbar surface
  (needs to be determined from assimilation
  models).

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Coefficients in mapping function

• The typical values for the coefficients are
• Hydrostatic
• a1.232e-3, b3.16e-3 c71.2e-3
• Wet delay
• a 0.583e-3 b1.402e-3 c45.85e-3
• Since coefficients are smaller for wet delay,
  this mapping function increases more rapidly at
  low elevation angles.
• At 0 degrees, hydrostatic mapping function is
  36. Total delay 82 meter

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Effects of atmospheric delay

• If atmospheric zenith delay not estimated, then
  when data is used to 10 degree elevation angle,
  error in height is 2.5 times zenith atmospheric
  delay error (see Herring, T. A., Precision of
  vertical position estimates from
  verylongbaseline interferometry, J. Geophys.
  Res., 91, 91779182, 1986. 
• A simple matlab program can reproduce these
  results
• Herring Kalman filter paper also discusses
  effects of process noise value in height estimate
  uncertainty.

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Parameterization of atmospheric delay

• Given the sensitivity of GPS position estimates
  to atmospheric delay, and that external
  calibration of the delay is only good to a few
  centimeters atmospheric zenith delays and often
  gradients are estimated high-precision GPS
  analyses.
• Parameterization is either Kalman filter or
  coefficients of piece-wise linear functions
  (GAMIT)

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Example using NCEP analysis field
Blue is GPS estimates of delay, red is NCEP
calculation
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Summary

• Atmospheric delays are one the limiting error
  sources in GPS
• In high precision applications the atmospheric
  delay are nearly always estimated
• At low elevation angles can be problems with
  mapping functions
• Spatial inhomogeneity of atmospheric delay still
  unsolved problem even with gradient estimates.
• Estimated delays are being used for weather
  forecasting if latency lt2 hrs.
• Material today
• Atmospheric structure
• Refractive index
• Methods of incorporating atmospheric effects in
  GPS